Evans 1st edition, exercise 7, chapter 5

Question

This is the exercise:

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Denote by $\Omega$ the square $\lbrace (x_1,x_2) \in \mathbb{R}^2 \vert \, \vert x_1 \vert < 1, \vert x_2 \vert<1 \rbrace$. Define $u(x)$ in this way:

$1-x_1$ if $\vert x_2 \vert < x_1$

$1+x_1$ if $\vert x_2 \vert < -x_1$

$1-x_2$ if $\vert x_1 \vert < x_2$

$1+x_2$ if $\vert x_1 \vert < -x_2$.

For which $p \in [1,\infty]$ does $u$ belong to $W^{1,p}(\Omega)$?

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I tried calculating the $\| \cdot \|_{W^{1,p}}$, but it is finite for all $p$. Is it correct or I made any mistake?

Answer 1

Yes, it is correct. You can easily see that $u \in W^{1,\infty}(\Omega)$ and since $\Omega$ is bounded, it follows that $u \in W^{1,p}(\Omega)$ for all $p \geq 1$.